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In Shannon formula below: $$R = W\log_2\left(1 +\frac{P_t\cdot H^2}{W N_0}\right)$$ why square the channel gain? When calculating the channel capacity, why do we need to square the channel gain?
Notation: $W$ is the bandwidth, $P_t$ is the transmit power, $H$ is the channel gain and $N_0$ is the thermal noise power density.
The Shannon channel capacity equation, as I remember is
$$\begin{align} R &= W \log_2 \left( 1 + \frac{S}{N} \right) \\ \\ &= W \log_2 \left( \frac{S+N}{N} \right) \\ \\ &= W \big( \log_2(S+N) \, - \, \log_2(N) \big) \\ \\ &= \frac{W}{6.02} \big( (S+N)_\text{dB} \, - \, (N)_\text{dB} \big) \\ \end{align}$$
where $S$ is the signal power and $N$ is the noise power (both over the same bandwidth $W$).
Noise power for a given bandwidth (thin enough that noise power density $N_0$ is considered constant is
$$ N = N_0 W $$
and the received signal power is the transmitted power, $P_t$, times the power gain of the channel (assuming both are constant over the bandwidth $W$). The power gain of the channel is the square of the gain in amplitude because instantaneous power is the square of instantaneous amplitude. So the received signal power is
$$ S = |H|^2 P_t $$
assuming all of the transmitted power is in the channel bandwidth $W$ and the channel gain, $H$, is constant over the bandwidth $W$.
